Gyárfás [1]} proved the conjecture for \(T=P_t\) : every \(P_t\) -free graph
\(G\)  has \(\chi (G)\le (t-1)^{\omega (G)-1}\) .
Note that this \(\chi \) -binding function is exponential in \(\omega (G)\) .
Esperet asked whether every \(\chi \) -bounded hereditary class is polynomially \(\chi \) -bounded.
Recently, this question was answered negatively by Briański, Davies and Walczak [2]}.
However, the question is still wide open for \(P_t\) -free graphs.
